论文标题
$ \ ell_1 $罚款功能的精确性，用于一类具有广义互补约束的数学程序
The Exactness of the $\ell_1$ Penalty Function for a Class of Mathematical Programs with Generalized Complementarity Constraints
论文作者
论文摘要
在具有广义互补性约束（MPGCC）的数学程序中，每对可变块之间施加了互补关系。 MPGCC包括具有互补性约束（MPCC）的传统数学程序。由于可行的区域，MPCC和MPGCC通常很难处理。 $ \ ell_1 $惩罚方法通常在计算中采用，开辟了一种规避难度的方式。然而，尚不清楚$ \ ell_1 $罚款功能的确切性，即，是否存在足够大的罚款参数，以便惩罚问题与原始罚款问题共享最佳解决方案。在本文中，我们考虑了一类具有多种目标功能的MPGCC。该问题类别在各个领域中找到了应用程序，例如，多体量子物理学中的多界数最佳传输问题以及网络传输中的定价问题。我们首先提供此类实例，其确切性的确切性无法由现有工具得出。然后，我们在相当温和的条件下建立精确性结果。我们的结果涵盖了MPCC现有的结果，并适用于多块上下文。
In a Mathematical Program with Generalized Complementarity Constraints (MPGCC), complementarity relationships are imposed between each pair of variable blocks. MPGCC includes the traditional Mathematical Program with Complementarity Constraints (MPCC) as a special case. On account of the disjunctive feasible region, MPCC and MPGCC are generally difficult to handle. The $\ell_1$ penalty method, often adopted in computation, opens a way of circumventing the difficulty. Yet it remains unclear about the exactness of the $\ell_1$ penalty function, namely, whether there exists a sufficiently large penalty parameter so that the penalty problem shares the optimal solution set with the original one. In this paper, we consider a class of MPGCCs that are of multi-affine objective functions. This problem class finds applications in various fields, e.g., the multi-marginal optimal transport problems in many-body quantum physics and the pricing problem in network transportation. We first provide an instance from this class, the exactness of whose $\ell_1$ penalty function cannot be derived by existing tools. We then establish the exactness results under rather mild conditions. Our results cover those existing ones for MPCC and apply to multi-block contexts.
